Fixed energy universality of Dyson Brownian motion

Landon, Benjamin; Sosoe, Philippe; Yau, Horng‐Tzer

doi:10.1016/j.aim.2019.02.010

Cited by 84 publications

(131 citation statements)

References 72 publications

(285 reference statements)

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“…(see [39] for existence of this derivative) and observe that Lemma 3.1. There exists C 0 > 0 such that with ϕ = e C0(log log N ) 2 , for any ν ∈ [0, 1], κ > 0 (small) and D > 0 (large), there exists N 0 (κ, D) so that for any N N 0 we have…”

Section: Coupling Of Determinantsmentioning

confidence: 90%

Gaussian fluctuations of the determinant of Wigner matrices

Bourgade¹,

Mody²

2019

Electron. J. Probab.

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We prove that the logarithm of the determinant of a Wigner matrix satisfies a central limit theorem in the limit of large dimension. Previous results about fluctuations of such determinants required that the first four moments of the matrix entries match those of a Gaussian [54]. Our work treats symmetric and Hermitian matrices with centered entries having the same variance and subgaussian tail. In particular, it applies to symmetric Bernoulli matrices and answers an open problem raised in [55]. The method relies on (1) the observable introduced in [10] and the stochastic advection equation it satisfies, (2) strong estimates on the Green function as in [12], (3) fixed energy universality [8], (4) a moment matching argument [53] using Green's function comparison [21].

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Section: Coupling Of Determinantsmentioning

confidence: 90%

Gaussian fluctuations of the determinant of Wigner matrices

Bourgade¹,

Mody²

2019

Electron. J. Probab.

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“…We apply Lemma 4.12 with the choice e 0 = E 1 : for any D > 0, for large enough N we have Part 3: Eigenvalues local statistics. We rely on fixed energy universality result for a matrix flow, from [22] (note that the constraint ω 0 > 2/3 below is probably not optimal but sufficient in our setting).…”

Section: 7mentioning

confidence: 99%

Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality

Bourgade

Yau

Yin

2020

Comm Pure Appl Math

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Consider N × N symmetric one-dimensional random band matrices with general distribution of the entries and band width W N 3/4+ε for any ε > 0. In the bulk of the spectrum and in the large N limit, we obtain the following results.(i) The semicircle law holds up to the scale N −1+ε for any ε > 0.(ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning their L ∞ norms are all simultaneously bounded by N − 1 2 +ε (after normalization in L 2 ) with overwhelming probability, for any ε > 0.(iv) Quantum unique ergodicity holds, in the sense that the local L 2 mass of eigenvectors becomes equidistributed with overwhelming probability. We extend the mean-field reduction method [4], which required W = Ω(N ), to the current setting W N 3/4+ε . Two new ideas are: (1) A new estimate on the "generalized resolvent" of band matrices when W N 3/4+ε . Its proof, along with an improved fluctuation average estimate, will be presented in parts 2 and 3 of this series [6,37]. (2) A strong (high probability) version of the quantum unique ergodicity property of random matrices. For its proof, we construct perfect matching observables of eigenvector overlaps and show they satisfying the eigenvector moment flow equation [7] under the matrix Brownian motions.

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“…Then for any t with N ε t N −ε L 2 the bulk universality of H t around E 0 holds both in the sense of correlation functions at fixed energy (6.3.18) and in sense of gaps (6.3.19). Theorem 6.3.20 in this general form appeared in [63] (gap universality) and in [62] (correlation functions universality at fixed energy). These ideas have been developed in several papers.…”

Section: Universality Of Local Eigenvalue Statisticsmentioning

confidence: 95%

The matrix Dyson equation and its applications for random matrices

Erdős¹

2019

Random Matrices

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These lecture notes are a concise introduction of recent techniques to prove local spectral universality for a large class of random matrices. The general strategy is presented following the recent book with H.T. Yau [45]. We extend the scope of this book by focusing on new techniques developed to deal with generalizations of Wigner matrices that allow for non-identically distributed entries and even for correlated entries. This requires to analyze a system of nonlinear equations, or more generally a nonlinear matrix equation called the Matrix Dyson Equation (MDE). We demonstrate that stability properties of the MDE play a central role in random matrix theory. The analysis of MDE is based upon joint works with J. Alt, O. Ajanki, D. Schröder and T.

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Fixed energy universality of Dyson Brownian motion

Cited by 84 publications

References 72 publications

Gaussian fluctuations of the determinant of Wigner matrices

Gaussian fluctuations of the determinant of Wigner matrices

Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality

The matrix Dyson equation and its applications for random matrices

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